Spahn directed object' (Rev #6, changes)

Definition (interval object)

Let $C$ be a closed monoidal (V-enriched for some $V$) homotopical category, let $*$ denote the tensor unit of $C$. Then the cospan category (same objects and cospans as morphisms) is $V$-enriched, too.

An interval object is defined to be a cospan $*\stackrel{a}{\rightarrow}I\stackrel{b}{\leftarrow}*$.

The pushout $I^{\coprod_2}$ of this diagram satisfies ${}_*[ I,I^{\coprod_2}]_*\simeq$ is contractible (see co-span for this notation).

Definition (directed object)

Let $V$ be a monoidal category, let $C$ be $V$-enriched, closed monoidal homotopical category, let $*$ denote the tensor unit of $C$ which we assume to be the terminal object, let $*\stackrel{0}{\to} I\stackrel{1}{\leftarrow}$ denote the interval object of $C$, let $X$ be a pointed object of $C$. Let $D$ be the functor bifunctor$D:C\times C\to V$ D:C\to D:C\times C\to V, $(X,I)\mapsto [I,X]$ X\mapsto (X,I)\mapsto [I,X].

A direction in $C$ is defined to be a subfunctor $d$ of $D(I,-)$ D D(I,-) preserving the terminal object and satisfying the properties below. In this case $d X$ is called a direction for $X$. A global element of $dX$ is called a $d$-directed path in $X$ and their collection we denote by $ddp(X)$ . the The properties are:

(1) The $D$ image of every map $I\to *\to X$ factoring over the point is in $ddp(X)$.

(2) $\mathrm{<span><del\; class="diffmod">\; ddp</del><ins\; class="diffmod">\; f</ins></span>}<span><del\; class="diffmod">\; (</del><ins\; class="diffmod">\; ,</ins></span>\mathrm{<span><del\; class="diffmod">\; X</del><ins\; class="diffmod">\; g</ins></span>}<span><del\; class="diffmod">\; )</del><ins\; class="diffmod">\; \in </ins></span>\mathrm{dX}$ ddp(X) f,g\in dX is are closed called under to the be tensor a product. For$\alpha,\beta\in ddp(X)$composable pair , their if pushout$\alpha \overline{f}<span><del\; class="diffmod">\; \otimes </del><ins\; class="diffmod">\; \circ </ins></span>\beta 0=\overline{g}\circ 1$ \alpha\otimes \overline \beta f\circ 0=\overline g\circ 1 where the overlined letters denote the adjoints under the hom adjunction. The composition of a composable pair is called defined their bycomposition$f\circ g:=\underline{\overline{f}\bullet\overline{g}}$ . Then the condition is that$ddp(X)$ is closed under composition.

A (3) Ifdirected object$\tau\in hom (I,I)$ is and defined to be a pair${}_{d}fX\in :\mathrm{ddp}=(X,\mathrm{dX})$ {}_d f\in X:=(X,dX) ddp(X) consisting then of an object$X\underline{\overline{f}\circ \tau }\in \mathrm{dX}$ X \underline{\overline{f}\circ \tau}\in dX of where the underlined term denotes the adjoint (in the other direction) under the hom adjunction.$C$ and a direction $dX$ for $X$.

A (4)morphism of directed objects$ddp(X)$ is a functor$f:(X,dX)\to (Y,dy)$ is defined to be a pair $(f,df)$ making